Use two distinct real roots to restrict a parameter

The numbers $x$ and $y$ satisfy the equation

$4x^2+16xy+y^2+16x+14y+13=0$.

If a real value of $x$ is substituted, the equation becomes a quadratic equation in $y$. Given that two distinct real values of $y$ may be found from this equation, show that

$5x^2+8x+3>0$,

and hence find the set of possible values of $x$.